Lecture 4 · Slides L4 · ABC ch. 3 to 5 · about 55 min

Macroeconomic Concepts: Labour, Saving and Investment

The three markets behind the long-run model: how the labour market sets employment and full-employment output, how households split income between consumption and saving, and how firms decide how much capital to hold.

By the end you can

Derive labour demand from MPN = w and explain what shifts labour supply
Define full-employment output and the natural rate of unemployment
Predict how income, wealth, the interest rate and fiscal policy affect desired saving
Compute the user cost of capital, the desired capital stock and gross investment
Use Tobin's q and the goods-market condition S = I

Production and productivity

This lecture follows Abel, Bernanke and Croushore (ABC), who write labour as $N$ rather than $L$ :

Production function (ABC notation)Formula sheet →

Y = A\,F(K, N)

$A$: productivity (technology and management)
$K$: capital
$N$: labour

Use it when you work in the ABC chapters. It is the same function as Mankiw's F(K, L), with productivity written in front.

Factors of production are capital, labour and others such as raw materials, land and energy. How much each factor produces depends on technology and management, summarised by $A$ . The Cobb-Douglas results from Lecture 3 still hold: labour’s share is $1 - \alpha$ , capital’s share is $\alpha$ .

The lecture breaks output per person into three drivers:

\text{Output per capita} = \text{productivity} \times \text{quality of human capital} \times \text{capital per worker}

Human capital is where the lecture points to Türkiye’s weakness. In the 2022 PISA survey of 15-year-olds in 81 countries, Türkiye ranked 39th in mathematics, 34th in science and 36th in reading, below the average. Compare top-100 universities: USA 34, Germany 9, China 7, Sweden 5, Switzerland 5, South Korea 3, Singapore 2.

The demand for labour

A firm thinks at the margin: what does one more worker cost and what does that worker add?

If the real wage $w$ is below $MPN$ , hiring one more worker raises profit.
If $w$ is above $MPN$ , that worker lowers profit.

So profit is highest where:

Labour demandFormula sheet →

w = MPN

$w$: real wage W/P
$MPN$: marginal product of labour

Use it when you derive the labour demand curve. Because MPN falls as N rises, the demand curve slopes down.

The labour demand curve is the MPN curve. Anything that raises the MPN at every $N$ shifts labour demand to the right: a beneficial supply shock (higher $A$ ) or a larger capital stock (each worker has more machines).

The supply of labour

Each person chooses between income and leisure. Working another day costs leisure but brings income to spend on consumption. Keep working extra days while the benefit exceeds the cost.

A rise in the real wage has two opposing effects:

Substitution effect: each hour of work pays more, so leisure is more expensive. Work more.
Income effect: you are richer for the same hours, so you can afford more leisure. Work less.

Aggregate labour supply slopes upward: when the real wage rises, some people work more hours (the intensive margin) and others enter the labour force (the extensive margin).

An increase in	shifts labour supply	because
Wealth	left	people can afford more leisure
Expected future real wage	left	future income makes more leisure affordable today
Working-age population	right	more potential workers
Participation rate	right	more people want to work

Labour market equilibrium

Where supply meets demand, the market sets the real wage and full-employment employment $\bar N$ . Put $\bar N$ into the production function:

Full-employment outputFormula sheet →

\bar{Y} = A\,F(K, \bar{N})

$\bar{N}$: equilibrium (full-employment) level of employment
$\bar{Y}$: full-employment or potential output

Use it when a question asks how a shock changes potential output. It moves if A, K or the labour market equilibrium changes.

Unemployment

Labour market ratiosFormula sheet →

u = \frac{U}{LF} \qquad \text{participation} = \frac{LF}{\text{adult pop.}} \qquad \text{employment ratio} = \frac{E}{\text{adult pop.}}

$LF$: labour force = E + U
$E$: employed
$U$: unemployed

Use it when a question gives numbers of people and asks for any of the three rates.

Unemployment comes in three kinds:

Frictional: workers and firms are different, so matching them takes time and search.
Structural: long-term, chronic unemployment that exists even outside recessions. Causes include a lack of skills and the slow reallocation of workers out of shrinking industries or depressed regions.
Cyclical: the extra unemployment during recessions.

Natural and cyclical unemploymentFormula sheet →

\bar{u} = \text{frictional} + \text{structural} \qquad \text{cyclical} = u - \bar{u}

$\bar{u}$: natural rate of unemployment, the rate at full employment
$u$: actual rate

Use it when a question asks how far the economy is from full employment in unemployment terms.

Consumption and saving

Income is either consumed or saved: $Y = C + S$ . Saving can be negative (dissaving) when $C > Y$ .

Saving is a trade-off between consumption today and tomorrow. One unit of consumption today costs $1 + r$ units of consumption next year, since you could have saved it at the real interest rate $r$ . People have a consumption-smoothing motive: they prefer a fairly even pattern of consumption over time.

How desired saving responds:

Current income rises: consumption and saving both rise. The MPC is between 0 and 1, so $C^d$ rises less than $Y$ and $S^d$ rises.
Expected future income rises: people consume more now, so current saving falls.
Wealth rises: some is consumed now, so saving falls.
Real interest rate rises: the substitution effect raises saving (the return is higher). The income effect lowers saving for a saver (less saving is needed to hit a target) but raises it for a borrower (who is poorer). Evidence suggests aggregate saving rises slightly.

Desired national savingFormula sheet →

S^d = Y - C^d - G

$C^d$: desired consumption
$G$: government purchases

Use it when you need the effect of fiscal policy or income changes on national saving.

Government purchases: a temporary rise in $G$ financed by taxes lowers after-tax income and so $C^d$ , but by less than $G$ rises. So $S^d$ falls. Lump-sum tax cut today, with higher taxes later: if people see the future taxes coming, they save the whole tax cut and $S^d$ does not change; if they do not, they consume part of it and $S^d$ falls.

Investment

Investment is small relative to GDP, but it swings sharply over the business cycle and drives long-run growth, so it deserves its own theory.

The desired capital stock

A firm holds the desired capital stock: the amount of capital that maximises expected profit. The benefit of one more unit of capital is its future marginal product, $MPK^f$ , since new capital takes time to install. The cost is the user cost of capital:

User cost of capitalFormula sheet →

uc = r\,p_K + d\,p_K = (r + d)\,p_K

$r$: real interest rate
$d$: depreciation rate
$p_K$: real price of a unit of capital

Use it when you need the cost side of the investment decision. r p_K is the interest cost (or forgone interest); d p_K is the wear and tear.

Desired capital stockConditionFormula sheet →

MPK^f = uc

$MPK^f$: expected future marginal product of capital

Use it when you find how much capital the firm wants. If MPK^f is above uc, add capital; if below, reduce it.

From desired capital to investment

The capital stock changes through two channels: new capital (gross investment) and depreciation. Net investment is the change in the capital stock:

Gross investmentFormula sheet →

I_t = \underbrace{K_{t+1} - K_t}_{\text{net investment}} + \underbrace{dK_t}_{\text{replacement}} = K^* - K_t + dK_t

$K^*$: desired capital stock
$K_t$: current capital stock
$d$: depreciation rate

Use it when a question gives the desired and current capital stock and asks for this period's investment, assuming firms reach K* in one period.

An increase in	desired investment	because
Real interest rate $r$	falls	user cost rises
Effective tax rate	falls	tax-adjusted user cost rises
Expected future $MPK$	rises	desired capital stock rises

Tobin’s q

Tobin's qFormula sheet →

q = \frac{V}{p_K K}

$V$: stock-market value of the firm
$p_K K$: replacement cost of its capital

Use it when a question links the stock market to investment. q above 1 means the market values capital above its cost: invest more. q below 1: do not invest.

A stock-market boom raises $V$ , raises $q$ and encourages investment. The logic matches the user-cost approach: a higher expected $MPK^f$ raises future earnings and so $V$ ; a lower real interest rate makes stocks more attractive than bonds and raises $V$ ; a cheaper $p_K$ raises $q$ directly.

Goods market equilibrium

In a closed economy the goods market clears when desired saving equals desired investment, and the real interest rate adjusts to make that happen:

Goods market equilibriumFormula sheet →

Y = C^d + I^d + G \quad\Longleftrightarrow\quad S^d = I^d

$S^d$: desired national saving
$I^d$: desired investment

Use it when you find the equilibrium real interest rate in the ABC model.

In an open economy desired saving and investment need not match. The gap is net exports, which equals net foreign lending and the current account balance:

S^d - I^d = NX

If $S^d > I^d$ the country lends abroad and $NX > 0$ ; if $S^d \lt I^d$ it borrows and $NX \lt 0$ . In a small open economy residents borrow and lend at the world real interest rate $r_w$ , which they cannot influence. At $r_w$ , desired saving and investment are read off their curves and the gap is the current account.

Summary and review

ABC notation: $Y = AF(K, N)$ . Labour demand: $w = MPN$ .
Labour supply: substitution effect (work more) vs income effect (work less). Temporary wage rise raises supply; permanent may lower it.
$\bar Y = AF(K, \bar N)$ is full-employment output.
$\bar u$ = frictional + structural; cyclical unemployment $= u - \bar u$ .
Saving rises with current income, falls with expected future income and wealth, probably rises slightly with $r$ .
$S^d = Y - C^d - G$ : higher $G$ lowers national saving.
$uc = (r + d)p_K$ ; desired capital where $MPK^f = uc$ ; $I = K^* - K + dK$ .
Tobin’s $q = V/(p_K K)$ : invest if $q > 1$ .
Closed: $S^d = I^d$ . Open: $S^d - I^d = NX$ .